Problem 74 - Calculus Early Transcendentals

Question

Evaluate the following limits or state that they do not exist. $$\lim _{\theta \rightarrow 0} \frac{\frac{1}{2+\sin \theta}-\frac{1}{2}}{\sin \theta}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit: \(\lim_{\theta\rightarrow0} \frac{\frac{1}{2+\sin\theta}-\frac{1}{2}}{\sin\theta}\) Answer: The limit of the given expression as \(\theta\) approaches 0 is \(\frac{-1}{4}\).

Step 1: Simplify the given expression

Simplify the fraction within the limit expression: $$\frac{\frac{1}{2+\sin\theta}-\frac{1}{2}}{\sin\theta} = \frac{1}{\sin\theta} \cdot \left(\frac{1}{2+\sin\theta}-\frac{1}{2}\right)=\frac{2-(2+\sin\theta)}{\sin\theta(2+\sin\theta)(2)}$$ Now simplify the numerator: $$\frac{2-(2+\sin\theta)}{\sin\theta(2+\sin\theta)(2)}=\frac{-\sin\theta}{\sin\theta(2+\sin\theta)(2)}$$

Step 2: Cancel out common factors

Cancel out the common factor \(\sin\theta\) in the numerator and denominator: $$\frac{-\sin\theta}{\sin\theta(2+\sin\theta)(2)}=\frac{-1}{(2+\sin\theta)(2)}$$

Step 3: Evaluate the limit

Now, apply the limit: $$\lim_{\theta\rightarrow0}\frac{-1}{(2+\sin\theta)(2)}=\frac{-1}{(2+\sin(0))(2)}=\frac{-1}{(2+0)(2)}=\frac{-1}{4}$$ So, the limit exists and it is equal to \(\frac{-1}{4}\).

Key Concepts

Trigonometric Limits

In calculus, trigonometric limits often involve functions like sine, cosine, and tangent. These functions are essential in determining behavior near specific points, like zero. Take, for example, the limit expression involving sine as noted in the exercise. When evaluating \[\lim _{\theta \rightarrow 0} \frac{\frac{1}{2+\sin \theta}-\frac{1}{2}}{\sin \theta}\], the sine function appears in both the numerator and denominator.

To tackle trigonometric limits:

• First, simplify the expression as much as possible. Look to factor or cancel out common terms.
• Convert the problem into a more familiar form, especially when sine, cosine, or tangent is involved.

Additionally, understanding the properties of sine and cosine at small angles, such as \(\sin(0) = 0\), helps. This allows you to evaluate limits confidently by predicting the behavior of the expression as \(\theta\) approaches zero, often simplifying the process. Recognizing indeterminate forms is also crucial here, which leads to the next important concept.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus for solving limits where direct substitution leads to an indeterminate form, like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). It involves taking derivatives of the numerator and denominator until an indeterminate form is resolved.

Applying this rule may reveal the true behavior of the fraction as the variable approaches a specific point. To use L'Hôpital's Rule:

• First, verify that the limit results in an indeterminate form.
• If valid, differentiate the numerator and the denominator separately.
• Then, re-evaluate the limit with these new expressions.

Remember, while not used directly in the provided exercise, it's useful in situations with similar rational expressions that reach an indeterminate form initially. Keeping this rule in your toolkit can help streamline the process of handling complex limits.

Indeterminate Forms

When dealing with limits, you’ll often encounter indeterminate forms. These are expressions where direct evaluation doesn't immediately reveal a limit's value. The most common indeterminate forms are \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\).

Consider the initial fraction \[\lim _{\theta \rightarrow 0} \frac{\frac{1}{2+\sin \theta}-\frac{1}{2}}{\sin \theta}\]. Direct substitution of \(\theta = 0\) results in \(\frac{0}{0}\), making it an indeterminate form.

To resolve indeterminate forms:

• Attempt algebraic manipulation, such as factorization or simplification.
• Use tools like L'Hôpital's Rule where differentiation can resolve the form efficiently.

Indeterminate forms indicate potential flexibility in resolving limits. Recognizing them helps determine the best strategy for a solution, whether through algebraic means, trigonometric identities, or calculus tools like L'Hôpital's Rule. Understanding these concepts leads to solving limits efficiently and correctly.